## Abstract and Applied Analysis

### A Two-Grid Method for Finite Element Solutions of Nonlinear Parabolic Equations

#### Abstract

A two-grid method is presented and discussed for a finite element approximation to a nonlinear parabolic equation in two space dimensions. Piecewise linear trial functions are used. In this two-grid scheme, the full nonlinear problem is solved only on a coarse grid with grid size $H$. The nonlinearities are expanded about the coarse grid solution on a fine gird of size $h$, and the resulting linear system is solved on the fine grid. A priori error estimates are derived with the ${H}^{1}$-norm $O(h+{H}^{2})$ which shows that the two-grid method achieves asymptotically optimal approximation as long as the mesh sizes satisfy $h=O({H}^{2})$. An example is also given to illustrate the theoretical results.

#### Article information

Source
Abstr. Appl. Anal., Volume 2012, Special Issue (2012), Article ID 391918, 11 pages.

Dates
First available in Project Euclid: 5 April 2013

https://projecteuclid.org/euclid.aaa/1365168880

Digital Object Identifier
doi:10.1155/2012/391918

Mathematical Reviews number (MathSciNet)
MR2975310

Zentralblatt MATH identifier
1253.65159

#### Citation

Chen, Chuanjun; Liu, Wei. A Two-Grid Method for Finite Element Solutions of Nonlinear Parabolic Equations. Abstr. Appl. Anal. 2012, Special Issue (2012), Article ID 391918, 11 pages. doi:10.1155/2012/391918. https://projecteuclid.org/euclid.aaa/1365168880

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